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Questions tagged [automorphism-group]

For finding, constructing and proving results about automorphisms and applying them into different contexts. An automorphism is an isomorphism from an object to itself, and collectively they form a group under composition of mappings. Sometimes the only automorphism is the trivial one (the identity map), but often structures will have interesting/non-trivial automorphisms.

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235 views

Are all verbal automorphisms inner power automorphisms?

Suppose $G$ is a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Lets call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, ...
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69 views

What can we say about the equivalence relation defined such that $G \sim H \Longleftrightarrow Aut(G) \simeq Aut(H)$?

Given two finite graphs $G, H$, I say that $G \sim H$ if and only if $Aut(G) \simeq Aut(H)$. We define $[G] = \{H \mid Aut(G) \simeq Aut(H) \}$ for any graph $G$. Due to theorems by Erdos and others ...
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30 views

What is the smallest poset with automorphism group $C_n$?

I've recently been interested in finding small finite posets (and thereby finite $T_0$ topologies) with a given automorphism group. I came upon the paper of Barmak and Minian in which they provide an ...
4
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59 views

Example of when $E^{H} \neq E^{\text{Aut}(E/E^H)}$?

Let $E/F$ be a field extension of $F$, and let $H \leq \text{Aut}(E/F)$ be a subgroup of the automorphisms of $E$ fixing $F$. If $H$ is finite, then it is well-known that $H = \text{Aut}(E/E^H)$ and ...
4
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113 views

For any sets $X$ & $Y$ can the group of automorphisms for the digraph $G(X,Y)=(X\cup Y,X\times Y)$ be express in terms of symmetric groups?

For any sets $X,Y\neq \emptyset$ if we define a permutation group $G(X,Y)$ on the set $X\cup Y$ as follows: $$G(X,Y)=\left\{\sigma\in \text{Sym}(X\cup Y):\forall x,y\left[(x,y)\in X\times Y\iff (\...
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109 views

Automorphisms of $\mathbb{A}^1_R$

When $R$ is an integral domain, the automorphisms of the affine line are all of the form $X \mapsto aX + b$ with $a \in R^\times$ and $b \in R$; the proof is the same as in the case of $R$ a field, ...
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$G$ non-centerless group. How is $Z(\operatorname{Aut}(G))$ made?

Let $G$ be a (possibly infinite) non-centerless group, i.e. such that $Z(G) \ne \lbrace e \rbrace$. Left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \...
3
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44 views

Automorphism of an open subset with completement of codimension $2$

Let $\mathbb P^n=\mathbb {CP^n}$, I guess the following is true: Let $D\subset \mathbb P^n$ be a closed subscheme of codimension $2$. Then every automorphism of $\mathbb P^n-D$ is linear, i.e. ${...
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51 views

Automorphism group of $\mathbb Z_2^3$

I am trying to find $\text{Aut}(\mathbb Z_2^3)$ and express it in terms of familiar groups and the direct and/or semi direct product. Here's what I have so far: I know that the set of generators $A:=\...
3
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37 views

Outer automorphism groups of finite groups

Is it known which finite groups are isomorphic to the outer automorphism group of some finite group? If not, what can be said about such groups - that is, is there any known class of finite groups ...
3
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126 views

Order of an Automorphism

Let $H$ be the group of integers mod $p$, under addition, where $p$ is prime. Suppose that $n$ is an integer $1 \leq n \leq p$, and let $G$ be the group $H \times H \times \dots \times H$ ($n$ times). ...
3
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118 views

Outer automorphisms of a connected Lie group

A compact, connected Lie group $G$ is finitely covered by a group of the form $T \times K$, where $T$ is a torus and $K$ is simply-connected. I am under the impression the outer automorphism group $$\...
3
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63 views

Automorphism group of $(\Bbb Z\times\Bbb Z) \rtimes \Bbb Z$

Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Lets say the generators of ...
3
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275 views

Characteristic subgroups of Aut(G)

For a group $G$, let Aut$(G)$ denote the group of all the automorphisms of $G$. Consider the following subgroups of Aut$(G)$: Inn$(G)=$the group of inner automorphisms of $G$ IA$(G)=$ the group of ...
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The automorphism group of the countable atomless Boolean algebra does not have ample generics

I was told that the automorphism group of the countable atomless Boolean algebra does not have ample generics. I assume that one would show this by using the Fraisse-theoretic characterizations of ...
2
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40 views

A question about groups, that are isomorphic to the automorphism groups of their cycle graphs

Suppose $G$ is a finite group. Let’s call an element $g \in G$ primitive, iff $\forall h \in G$, such that $\exists n \in \mathbb{N}, h^n = g$ it is true, that $\langle h \rangle = \langle g \rangle$. ...
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15 views

Internal Semidirect with Factors Isomorphic to “Outside” Groups

Here is a conjecture of mine: If $G$ is the internal semidirect product of $N \unlhd G$ and $Q \le G$, and $\phi_1 : N' \to N$ and $\phi_2 : Q' \to Q$ are isomorphisms, then there is some $\theta :...
2
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33 views

automorphism group of splitting field of cubic polynomial over $\mathbb{Q}$

The question is to compute the isomorphism class of automorphism group of the splitting field over $\mathbb{Q}$ of $x^3 - 6x + 2$. I know that this polynomial is irreducible over $\mathbb{Q}$ by ...
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54 views

Burnside groups : a few questions

Let $G=B(n,e)=F_n/\langle\langle F_n^e\rangle\rangle$ be the Burnside group on $n$ generators with exponent $e$, i.e. the quotient of the free group on $n$ generators $F_n$ by the normal subgroup ...
2
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44 views

Find the order of $\operatorname {Aut}(C_p^{\oplus n})$

Let $C_p$ be a cyclic group of prime order $p$. Find the number of the automorphism group of $C_p^{\oplus n}$. Is this approach correct? $C_p$ is isomorphic to $(\mathbb Z_p,+)$. The automorphism ...
2
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65 views

Inner Automorphisms of a Lie Subalgebra

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}\subseteq\mathfrak{g}$ a subalgebra. If $\tau$ is an inner automorphism of $\mathfrak{h}$, can we always lift $\tau$ to an inner automorphism of $\...
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77 views

Automorphism of $U(16)$

I wanted to find Automorphism group of $U(16)$. I know that $U(16) \approx Z_2 \times Z_4$ .Which is not cyclic. Actually I can give automorphism in case of cyclic group using generator of that group ...
2
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37 views

Is $*:U(R)\times R^+\longrightarrow R^+,\ (g,a)\mapsto g*a:=g\cdot a$ surjective?

Let $R=(R,+,\cdot)$ be a ring with $1_R$. We will write $U(R)=(U(R),\cdot)$ the multiplicative group of units of this ring and $R^+=(R,+)$ the additive group of this ring. The group $U(R)$ acts in the ...
2
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48 views

A weird automorphism

Consider the polynomial $f(t) = t^5 -4t+2$ over $\mathbb{Q}$. By Eisenstein's criterion, this is irreducible. By graphing this function, we see that it has 3 real roots, hence 2 (distinct, because ...
2
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55 views

Relating the automorphisms of $P$ and those of $P/Φ(P)$

Below is part of an exercise in Section 6.1 of Abstract Algebra: Dummit and Foote. 26. (e) Let $p$ be a prime. Let $P$ be a finite $p$-group. Let $\sigma$ be any automorphism of $P$ of prime order $...
2
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52 views

Centralizer of an outer automorphism

Let $G$ be a finite centerless group, which we identify with $\operatorname{Inn}(G)$, and note $A := \operatorname{Aut}(G)$. Suppose further that $[A:G] = 2$, and take a non-inner automorphism $\alpha ...
2
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38 views

Can every group be embedded normally in a group such that its automorphisms are all conjugations?

Let $H$ be a group. Can $H$ be embedded into a bigger group $G$ such that $H\triangleleft G$ and for every $\varphi\in\text{Aut}(H)$ there exists $g_\varphi\in G$ with $\varphi(h)=g_\varphi hg_\varphi^...
2
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118 views

How to find Automorphism group of $\mathbb Z_q \times\mathbb Z_q$ where $q$ is a prime.

I know that automorphism group of $\mathbb Z_q$ is $(q-1)$ because the group is cyclic and its generator's image determines the automorphism and we have exactly $(q-1)$ choices. For $\mathbb Z_q \...
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18 views

Let $M$ be a finite model of a language where nonlogical symbols are only constants and symbols of $1-$aria relation.

Let Aut$ (M)$ be the automorphism group ($M$ isomorphisms in and of itself). Show That there are natural numbers $n_1,. . . , n_k$ such that the number of elements of Aut $(M)$ is $n_1! · · · n_k!$. ...
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36 views

Finding the automorphism group of a graph

Let me denote the graph in the picture by $\Gamma$ and I will refer to the points as numbers $1-9$. I need to find Aut($\Gamma$). Looking at this graph, it seems that there can be a permutation on ...
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22 views

Preserving relations and operations in an automorphism

An automorphism is an isomorphism of an algebraic structure with itself. Let's review an isomorphism $i$ of two linearly ordered groups $G(+,<)$ and $F(*,<)$. $i$ is linear order-preserving ...
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29 views

Inner Automorphism using Transpositions

Test question that I got destroyed by. Thought about it all weekend and came up with an explanation. Today, scores back, and prof gave a very different explanation from the one I had in mind. But I'm ...
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41 views

Almost all trees have non-trivial automorphism group

In their paper Asymmetric Graphs Erdős and Rényi proved that almost all trees have non-trivial automorphism group. More specifically they showed that almost all trees contain at least one so-called ...
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38 views

Group Automorphisms of Norm 1 Groups in Quaternion Algebras

Let $A$ be a quaternion algebra over a field $K$ of characteristic zero. By the famous Skolem-Noether theorem, every $K$-algebra automorphism $\varphi \colon A \to A$ is of the form $$ \varphi(x) = ...
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35 views

Group of automorphisms of the disk fixing the boundary

I would like to know everything you know (group structure, dense or interesting subsets etc) about the group of diffeomorphisms $$\psi: \mathbb{D}^n \to \mathbb{D}^n$$ that such that $\psi|_{\partial ...
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13 views

Automorphism of subgroup generated by permutations

In group $S_5$ let's take the subgroup $H$ generated by permutations $a=(2\,3\,5)$ and $b=(3\,2\,5)$. Find AutH. So, I have found out that $ab=ba=e$ and $aa=b$ and $bb=a$. So $H$ is $\{e,a,b\}$. Is ...
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37 views

How to compute quickly the order of the automorphism group of $A_4$

I'm recently got interested by a series of very cool videos by Math Doctor Bob, specifically the ones about to compute automorphism groups. The one puzzling me a bit is this one: Automorphism of A4 ...
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188 views

Any group automorphism can preserve some nontrivial subgroups if the group is non-abelian

Question: Let $G$ be a finite group. If the group automorphism $\sigma$ satisfies that $\sigma(H)=H$ for some nontrivial subgroup $H$, we call $\sigma$ has the preserving property. Now, let $G$ be a ...
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0answers
37 views

Lifting finite order automorphism of power series ring

$\require{AMScd}$ Let $\mathcal{O}$ be an (finite) extension of the $p$-adic integers.. Denote by $R$ the power series ring $\mathcal{O} [[X_1, \dots, X_n]]$ and let $I$ be an ideal such that $R/I$ ...
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Actions of $C^*$-dynamical systems on primitive ideals

Reading about induced Systems of $C^*$-Algebras, I found this one statement that I can't figure out. Let $G$ be a compact Group and $(A,G,\alpha)$ a $C^*$-dynamical System, such that for some closed ...
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49 views

Explicit expressions of inner / outer automorphism of symplectic group

We know that one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G). In this way, the outer automorphism of symplectic group $Sp(n)$ (some people may denote it ...
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57 views

Explicit expressions of inner / outer automorphism of special orthogonal group SO(N)

I am aware the general statement on inner / outer automorphism of special orthogonal group SO($N$). Here I am trying to show them very explicitly. Consider SO(3) Lie algebra generators: $$ [X_i,X_j]=...
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0answers
74 views

Automorphism on finite group taking more than $3/4th $ elements going to inverse then G is abelian

I know this question is already answered but My argument is different I wanted t check that is I am giving right arrgument? Let G be the group and T is automorphism which send more than $3/4th$ ...
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0answers
27 views

The forms of $f,g \in k[x,y]$ such that $k[f,g,x]=k[x,y]$

Let $f=f(x,y), g=g(x,y) \in k[x,y]$ be two non-constant polynomials in two variables over a field of characteristic zero $k$. Assume that $k[f,g,x]=k[x,y]$ and that $Jac(f,g):=f_xg_y-f_yg_x \in k[x,y]...
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37 views

Extension of field automorphisms

In the case of finite extension of finite field extension it's easy to see that every automorpshim can be extended in a "good" way. Now, my question is : Is it possible also in the case of infinite ...
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0answers
30 views

Automorphisms of order 3 in $C_2$ X $C_2$ X $C_2$

I am trying to find out the number of automorphisms there are in $C_2$ $\times$ $C_2$ $\times$ $C_2$ $\cong <x,y,z|x^2=y^2=z^2=1>$. I only can come up with the one which moves $x \rightarrow y \...
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0answers
44 views

Explanation for anti-automorphism

https://books.google.si/books?id=NZHkBwAAQBAJ&pg=PA142&lpg=PA142&dq=complementary+bases+for+dual+space&source=bl&ots=fWIjy8AHzt&sig=t9hUYoov-mcPEsSMWGn71YLrABg&hl=ru&sa=...
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0answers
129 views

Why every automorphism of $S_3$ is inner Automorphism But This is not case with $S_6$?

I can show Inner Automorphism of $S_3$ are isomorphic to $S_3$ .Why every automorphism of $S_3$ is inner automorphism only? That is why there is no automorphism of $S_3$ exist that is not inner ...
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0answers
43 views

$\mathbb{Q}(i,\sqrt{2})$ is a splitting field of $f_{\mathbb{Q}}^{i+\sqrt{2}} $ and $\operatorname{Aut}(\mathbb{Q}(i,\sqrt{2})) \cong V_{4}$

So hi all, I'm trying to show that $\operatorname{Aut}(\mathbb{Q}(i,\sqrt{2}))$ is isomorphic to $V_4$ but I don't have enough intuition for this yet to solve it. Would you like to give me some advice?...
1
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0answers
21 views

Find a particular automorphism

Let $\Omega$ be $(\mathbb{R}^-\cup{0})^c$. Find an automorphism that has $1$ as a fixed point and it is not the identity. Could an inversion of the plane across the $xy$ axis be an automorphism that ...